The generalization of signal and image processing to surfaces entails filtering the normals of the surface, rather than filtering the positions of points on a mesh. Using a variational framework, smooth surfaces minimize the norm of the derivative of the surface normals -- i.e. total curvature. Penalty functions on the surface normals are computed using geometry-based shape metrics and minimized using gradient descent. This produces a set of partial differential equations (PDE). In this paper, we introduce a novel framework for implementing geometric processing tools for surfaces using a two step algorithm: (i) operating on the normal map of a surface, and (ii) manipulating the surface to fit the processed normals. The computational approach uses level set surface models; therefore, the processing does not depend on any underlying parameterization. Iterating this two-step process, we can implement geometric fourth-order flows efficiently by solving a set of coupled second-order PDEs. This paper will demonstrate that the framework provides for a wide range of surface processing operations, including edge-preserving smoothing and high-boost filtering. Furthermore, the generality of the implementation makes it appropriate for very complex surface models, e.g. those constructed directly from measured data.

Registration of 3D medical images consists in computing the “best” transformation between two acquisitions, or equivalently, determines the point to point correspondence between the images. Registration algorithms are usually based either on features extracted from the image (feature-based approaches) or on the optimization of a similarity measure of the images intensities (intensitybased or iconic approaches). Another classification criterion is the type of transformation sought (e.g. rigid or non-rigid). In this chapter, we concentrate on feature-based approaches for rigid registration, similar approaches for non-rigid registration being reported in another set of publication [35, 36]. We show how to reduce the dimension of the registration problem by first extracting a surface from the 3D image, then landmark curves on this surface and possibly landmark points on these curves. This concept proved its efficiency through many applications in medical image analysis as we will see in the sequel. This work has been for a long time a central investigation topic of the Epidaure team [2] and we can only reflect here on a small part of the research done in this area. We present in the first section the notions of crest lines and extremal points and how these differential geometry features can be extracted from 3D images. In Section 2, we focus on the different rigid registration algorithms that we used to register such features. The last section analyzes the possible errors in this registration scheme and demonstrates that a very accurate registration could be achieved.

Segmentation of a 3-dimensional (3D) polygonal mesh is a method of breaking the mesh down into \meaningful " connected subsets of vertices called regions. The method used here is based on the watershed segmentation scheme which appears prominently in the image segmentation literature and was later applied to the 3D segmentation problem. The watershed algorithm is based on a scalar value (curvature) assigned to every mesh vertex which purports to capture some essence of the local mesh shape. Accuracy of the curvature estimate at mesh vertices is critical to the quality of the resulting segmentation. The basis of this paper is one such watershed based 3D mesh segmentation scheme, described in [17], wherein two discrete curvature estimation schemes were used, i.e. the curvature was extracted directly from the mesh geometry. This paper considerably improves on the quality of segmentation in the original paper primarily by applying more accurate and robust curvature estimation techniques. The results are illustrated on data sets which are inherently noisy, such as the ones obtained from laser digitizers. 1

While vortex region quantities are Galilean invariant, most methods for extracting vortex cores depend on the frame of reference. We present an approach to extracting vortex core lines independently of the frame of reference by extracting ridge and valley lines of Galilean invariant vortex region quantities. We discuss a generalization of this concept leading to higher dimensional features. For the visualization of extracted line features we use an iconic representation indicating their scale and extent. We apply our approach to datasets from numerical simulations and experimental measurements.

Motivated by the goals of improving detection of low-contrast and narrow vessels and eliminating false detections at non-vascular structures, a new technique is presented for extracting vessels in retinal images. The core of the technique is a new likelihood ratio test that combines matchedfilter responses, confidence measures and vessel boundary measures. Matched filter responses are derived in scale-space to extract vessels of widely varying widths. A vessel confidence measure is defined as a projection of a vector formed from a normalized pixel neighborhood onto a normalized ideal vessel profile. Vessel boundary measures and associated confidences are computed at potential vessel boundaries. Combined, these responses form a 6-dimensional measurement vector at each pixel. A training technique is used to develop a mapping of this vector to a likelihood ratio that measures the "vesselness" at each pixel. Results comparing this vesselness measure to matched filters alone and to measures based on the Hessian of intensities show substantial improvements both qualitatively and quantitatively. The Hessian can be used in place of the matched filter to obtain similar but less-substantial improvements or to steer the matched filter by preselecting kernel orientations. Finally, the new vesselness likelihood ratio is embedded into a vessel tracing framework, resulting in an e#cient and e#ective vessel centerline extraction algorithm.

... about the shapes of objects and can be intuitively defined as curves on a surface along which the surface bends sharply. Our mathematical description of such surface creases is based on study of extrema of the principal curvatures along their curvature lines. On a smooth generic surface we define ridges to be the local positive maxima of the maximal principal curvature along its associated curvature line and ravines to be the local negative minima of the minimal principal curvature along its associated curvature line. The ridges and ravines are important for shape analysis and possess remarkable mathematical properties. For example, they correspond to the end points of shape skeletons. In this paper we derive formulas to detect the ridges and ravines on a surface given in implicit form. We also propose an algorithm of obtaining piecewise linear approximation of ridges and ravines as intersection curves of two implicit surfaces.

Abstract — We present an approach to analyze mixing in flow fields by extracting vortex and strain features as extremal structures of derived scalar quantities that satisfy a duality property: they indicate vortical as well as high-strain (saddletype) regions. Specifically, we consider the Okubo-Weiss criterion and the recently introduced MZ-criterion. While the first is derived from a purely Eulerian framework, the latter is based on Lagrangian considerations. In both cases high values indicate vortex activity whereas low values indicate regions of high strain. By considering the extremal features of those quantities, we define the notions of a vortex and a strain skeleton in a hierarchical manner: the collection of maximal 0D, 1D and 2D structures assemble the vortex skeleton; the minimal structures identify the strain skeleton. We extract those features using scalar field topology and apply our method to a number of steady and unsteady 3D flow fields. Index Terms — flow visualization, feature extraction, vortex core lines, strain features I.

We propose a new computational method for segmenting topological sub-dimensional point-sets in scalar images of arbitrary spatial dimensions. The technique is based on calculating the homotopy class defined by the gradient vector in a sub-dimensional neighborhood around every image point. This neighborhood is defined as the linear envelope spawned over a given sub-dimensional vector frame. In the simplest case where the rank of this frame is maximal, we obtain a technique for localizing the critical points, i.e. extrema and saddle points. We consider in particular the important case of frames formed by an arbitrary number of the first largest by absolute value principal directions of the Hessian. The method then segments positive and and negative ridges as well as other types of critical surfaces of different dimensionalities. The signs of the eigenvalues associated to the principal directions provide a natural labeling of the critical sub-sets.

Abstract—An automated method is presented for selecting optimal parameter settings for vessel/neurite segmentation algorithms using the minimum description length principle and a recursive random search algorithm. It trades off a probabilistic measure of image-content coverage against its conciseness. It enables nonexpert users to select parameter settings objectively, without knowledge of underlying algorithms, broadening the applicability of the segmentation algorithm, and delivering higher morphometric accuracy. It enables adaptation of parameters across batches of images. It simplifies the user interface to just one optional parameter and reduces the cost of technical support. Finally, the method is modular, extensible, and amenable to parallel computation. The method is applied to 223 images of human retinas and cultured neurons, from four different sources, using a single segmentation algorithm with eight parameters. Improvements in segmentation quality compared to default settings using 1000 iterations ranged from 4.7%–21%. Paired-tests showed that improvements are statistically significant @ H HHHSA. Most of the improvement occurred in the first 44 iterations. Improvements in description lengths and agreement with the ground truth were strongly correlated @ aHUVA. Index Terms—Image segmentation, minimum description length, optimization methods, segmentation evaluation. I.

A method is presented to extract space curves, defined by differential invariants, at increasing scales. The curves are considered as the intersection of two iso-surfaces in 3D, so their moving paths or orbits can be explicitly obtained in scale space as the intersection of two iso-surfaces in 4D. This method is based on a novel algorithm to search for iso-surfaces and their intersections in nD. The algorithm is a significant extension of the 3D Marching Lines algorithm with new orientation and implementation considerations. As a result of these considerations, the reconstructed iso-surfaces and their intersections can be proved to have good topological properties; moreover the implementation is quite straightforward. Specifically, a 4D extension has been implemented to follow spatial curves efficiently via scale. The algorithm automatically finds the connection order of singularities, so tracking remains reliable even if scale is not densely sampled. As an example, the develo...